bash log to file with timestamp and log type prefix - based on http://serverfault.com/a/310104/15215

log.sh

log() {
    while IFS= read -r line; do
        # prepend timestamp, first argument/second argument, and then input line
        echo "[$(date -u +'%Y-%m-%d %H:%M:%S,%3N') $1/$2] $line"
    done
}

script.sh

#!/bin/bash

# load the log function definition
. log.sh

# log everything output from this script to logfile with prefixes 
exec > >(log INFO $0 >> script.log)
exec 2> >(log ERROR $0 >> script.log)

echo "HUZZAH!"

cat /file/does/not/exist